nLab
pushout

The idea

In the category Set a ‘pushout’ is a quotient of the disjoint union of two sets. Given a diagram of sets and functions like this:

$$\begin{array}{ccccccccc}& & & & C& & & & \\ & & & f\swarrow & & \searrow g& & \\ & & A& & & & B\end{array}$$\array{ &&&& C &&&& \\ & && f \swarrow & & \searrow g && \\ && A &&&& B }

the ‘pushout’ of this diagram is the set $X$ obtained by taking the disjoint union $A+B$ and identifying $a\in A$ with $b\in B$ if there exists $x\in C$ such that $f(x)=a$ and $g(x)=b$ (and all identifications that follow to keep equality an equivalence relation).

This construction comes up, for example, when $C$ is the intersection of the sets $A$ and $B$, and $f$ and $g$ are the obvious inclusions. Then the pushout is just the union of $A$ and $B$.

Note that there are maps $i_A:A\to X$, $i_B:B\to X$ such that $i_A(a)=[a]$ and $i_B(b)=[b]$ respectively. These maps make this square commute:

$$\begin{array}{ccccccccc}& & & & C& & & & \\ & & & f\swarrow & & \searrow g& & \\ & & A& & & & B\\ & & & {}_{i_A}\searrow & & {\swarrow }_{i_B}& & \\ & & & & X& & & & \end{array}$$\array{ &&&& C &&&& \\ & && f \swarrow & & \searrow g && \\ && A &&&& B \\ & && {}_{i_A}\searrow & & \swarrow_{i_B} && \\ &&&& X &&&& }

In fact, the pushout is the universal solution to finding a commutative square like this. In other words, given any commutative square

$$\begin{array}{ccccccccc}& & & & C& & & & \\ & & & f\swarrow & & \searrow g& & \\ & & A& & & & B\\ & & & {}_{j_A}\searrow & & {\swarrow }_{j_B}& & \\ & & & & Y& & & & \end{array}$$\array{ &&&& C &&&& \\ & && f \swarrow & & \searrow g && \\ && A &&&& B \\ & && {}_{j_A}\searrow & & \swarrow_{j_B} && \\ &&&& Y &&&& }

there is a unique function $h:Y\to X$ such that

$$h{j}_A=i_A$$h j_A = i_A

and

$$h{j}_A=i_B.$$h j_A = i_B .

Since this universal property expresses the concept of pushout purely arrow-theoretically, we can formulate it in any category. It is, in fact, a simple special case of a colimit.

Definition

A pushout is a colimit of a diagram like this:

$$\begin{array}{ccccccccc}& & & & c& & & & \\ & & & f\swarrow & & \searrow g& & \\ & & a& & & & b\end{array}$$\array{ &&&& c &&&& \\ & && f \swarrow & & \searrow g && \\ && a &&&& b }

Such a diagram is called a span. If the colimit exists, we obtain a commutative square

$$\begin{array}{ccccccccc}& & & & c& & & & \\ & & & f\swarrow & & \searrow g& & \\ & & a& & & & b\\ & & & {}_{i_a}\searrow & & {\swarrow }_{i_b}& & \\ & & & & x& & & & \end{array}$$\array{ &&&& c &&&& \\ & && f \swarrow & & \searrow g && \\ && a &&&& b \\ & && {}_{i_a}\searrow & & \swarrow_{i_b} && \\ &&&& x &&&& }

and the object $x$ is also called the pushout. It has the universal property already described above in the special case of the category $\mathrm{Set}$.

Note that the concept of pushout is dual to the concept of pullback: that is, a pushout in $C$ is the same as a pullback in $C^{\mathrm{op}}$.